E-Courant Algebroids

Transcription

1 E-Courant Algebroids Zhuo Chen, Zhangju Liu and Yunhe Sheng Department of Mathematics and LMAM, Peking University, Beijing, , China Abstract A kind of generalized Courant algebroids is introduced associated to a vector bundle E, called an E-Courant algebroid, with the differential operator bundle DE as the target of its anchor. Such a new structure not only unifies several known structures such as Courant algebroids, Courant-Jacobi algebroids and omni-lie algebroids but also provides some interesting topics for the future studies. 1 Introduction In resent years, Courant algebroids are far and wide studied from several aspects and have been found many applications such as Manin pairs and moment maps [1], [3]; generalized complex structures [2], [10]; L -algebras and symplectic supermanifolds [25]; gerbes [28] as well as BV algebras and topological field theories [13], [26]. There are also two kinds of geometric structures related to Courant algebroids closely. First, the notion of Jacobi bialgebroids or, equivalently, generalized Lie bialgebroids was introduced in [8] and [14] respectively in order to generalize Dirac structures on Poisson manifolds to Jacobi manifolds. More general forms are generalized Courant algebroids [24] or Courant-Jacobi algebroids [9]. Another one is a so-called omni-lie algebroid given in [5] in order to characterize all possible Lie algebroid structures on a vector bundle E, which generalizes the notion of an omni-lie algebra introduced in [30] by defining some kind of algebraic structures on gl(v ) V for a vector space V. Such an algebra is not a Lie algebra but all possible Lie algebra structures on V can be characterized by its Dirac structures. The omni-lie algebra can be regarded as the linearization of the Courant algebroid structure on T M T M at a point and are studied from several aspects recently ([2], [15], [29]). In general, the features of Dirac structures of omni-lie algebroids are studied by the authors in [6]. In this paper, after analyzing some common points between Courant-Jacobi algebroids and omni-lie algebroids, we introduce a kind of generalized Courant algebroids associated to a vector bundle E, called an E-Courant algebroid, with the differential operator bundle DE as the target of its anchor. Such a new structure not only unifies the two structures mentioned above but also provides a number of interesting examples, i.e., the T M-Courant algebroid structure on the jet bundle of a Courant algebroid over M. An exact E-Courant algebroid can also be defined by a similar way for an exact Courant algebroid given in [28], where any exact Courant algebroid on T M T M comes from a twist of the standard one by a closed 3-form. This structure leads to twisted Poisson structures and is related to gerbes and topological sigma models. Notice that an important reason for this fact being true is because that any exact Courant algebroid has an isotropic splitting, i.e., both T M and T M are isotropic subbundles. Unfortunately, such a fact is not always true for an exact E-Courant algebroid if rank(e) 2. Therefore, for general cases, 0 MSC: Primary 17B65. Secondary 18B40, 58H05. Research partially supported by NSF of China and China Postdoctoral Science Foundation ( ). The third author is also supported by the governmental scholarship from China Scholarship Council. 1

2 we need to use the Leibniz cohomology, a noncommutative generalization of the derham cohomology ([20], [21]). The paper is organized as follows. In Section 2, first we introduce the notion of E-Courant algebroids. Then, after showing several examples, we prove that the jet bundle JC of a Courant algebroid C over M admits a natural T M-Courant algebroid structure. In Section 3, we discuss some basic properties of the E-dual pair of Lie algebroids in order to introduce the notion of E-Lie bialgebroids later. Another purpose is to construct a long exact sequence of the skew-symmetric jet bundles. As an immediate application of this sequence, in Section 4, we fix the automorphism group and the possible twists of omni- Lie algebroids. In Section 5, we study exact E-Courant algebroids, which are isomorphic to DE JE as vector bundles. We prove that any exact E-Courant algebroid with an isotropic splitting is isomorphic to the omni-lie algebroid. In general, an exact E-Courant algebroid is associated with a 2-cocycle in the Leibniz cohomology of Γ(DE) with coefficients in Γ(JE), which can also be considered as a 3-cocycle in the Leibniz cohomology of Γ(DE) with coefficients in Γ(E). In Section 6, we mainly study E-Lie bialgebroids, which is a generalization of generalized Lie bialgebroids. In Section 7, we extend the theory of Manin triples from Lie bialgebroids to E-Lie bialgebroids and give some examples. Acknowledgement: Z. Chen would like to give his warmest thanks to P. Xu for useful comments and the hospitality during his visit at PSU. Y. Sheng gives special thanks to Luuk Hoevenaars, Marius Crainic and Ieke Moerdijk for their help during his stay in Utrecht University. 2 E-Courant algebroids For a vector bundle E M, its gauge Lie algebroid DE with bracket [, ] D is just the gauge Lie algebroid of the frame bundle F(E), which is also called the covariant differential operator bundle of E (see [22, Example 3.3.4]). The corresponding Artiyah sequence is as follows: 0 gl(e) i DE j T M 0. (1) In [5], the authors proved that the jet bundle JE (see [7], [27] for more details about jet bundles) may be considered as an E-dual bundle of DE, i.e., JE = {ν Hom(DE, E) ν(φ) = Φ ν(1 E ), Φ gl(e)} Hom(DE, E). Associated with the jet bundle JE, the jet sequence of E is given by: 0 Hom(T M, E) e JE p E 0. (2) The operator d : Γ(E) Γ(JE) is given by: du(d) := d(u), u Γ(E), d Γ(DE). An important formula which will be often used is d(fx) = df X + fdx, X Γ(C), f C (M). (3) For a vector bundle K over M and a bundle map ρ : K DE, we denote the induced E-adjoint bundle map by ρ, i.e., ρ : Hom(DE, E) Hom(K, E), ρ (ν)(k) = ν(ρ(k)), k K, ν Hom(DE, E). The notion of Leibniz algebras was introduced by Loday [19], [20] (also see [11]). A Leibniz algebra g is an R-module, where R is a commutative ring, endowed with a linear map [, ] : g g g satisfying With the above notations, we have [g 1, [g 2, g 3 ]] = [[g 1, g 2 ], g 3 ] + [g 2, [g 1, g 3 ]], g i g. 2

3 Definition 2.1. An E-Courant algebroid is a quadruple (K, (, ) E, [, ] K, ρ), where K is a vector bundle over M such that (Γ(K), [, ] K ) is a Leibniz algebra, (, ) E : K K E is a nondegenerate E-valued inner product, which induces an embedding: K Hom(K, E), and the anchor ρ : K DE is a bundle map such that the following properties hold for all X, Y, Z Γ(K): EC-1) ρ[x, Y ] K = [ρ(x), ρ(y )] D ; EC-2) [X, X] K = ρ d (X, X) E ; EC-3) ρ(x) (Y, Z) E = ([X, Y ] K, Z) E + (Y, [X, Z] K ) E ; EC-4) ρ (JE) K, i.e., (ρ (µ), X) E = 1 2 µ(ρ(x)), µ JE; EC-5) ρ ρ = 0. Remark 2.2. If the E-valued pairing (, ) E : K K E is surjective, we can obtain the properties (EC-4) and (EC-5) from the property (EC-2). In particular, if E is a line bundle, any nondegenerate E-valued pairing (, ) E is surjective. Lemma 2.3. For any X, Y Γ(K) and f C (M), we have [X, fy ] K = f[x, Y ] K + (j ρ(x)f)y, (4) [fx, Y ] K = f[x, Y ] K (j ρ(y )f)x + 2ρ (df (X, Y ) E ). (5) Proof. By the property (EC-3), for all X, Y, Z Γ(K) and f C (M), we have ([X, fy ] K, Z) E + (fy, [X, Z] K ) E = ρ(x) (fy, Z) E = j ρ(x)(f) (Y, Z) E + fρ(x) (Y, Z) E = j ρ(x)(f) (Y, Z) E + f ([X, Y ] K, Z) E + f (Y, [X, Z] K ) E. The nondegeneracy of the pairing (, ) E yields that By the property (EC-2), we have [X, fy ] K = j ρ(x)(f)y + f[x, Y ] K. [X, fy ] K + [fy, X] K = 2ρ d(f (X, Y ) E ) = 2fρ d (X, Y ) E + 2ρ (df (X, Y ) E ). Substitute [X, fy ] K by (4) and apply the property (EC-2) again, we obtain (5). For a subbundle L K, denote by L K as L = {e K (e, l) E = 0, l L}. Definition 2.4. A Dirac structure of an E-Courant algebroid (K, (, ) E, [, ] K, ρ) is a subbundle L K which is closed under the bracket [, ] K and satisfies L = L. Evidently, L = L implies that L is maximal isotropic with respect to the E-valued pairing (, ) E. In general, by the condition that L is maximal isotropic with respect to (, ) E, we can not obtain the equality that L = L as showing follows. Example 2.5. Let K be R 3 with the standard basis e 1, e 2, e 3. The R-valued pairing (, ) R is given by (e 1, e 3 ) R = (e 2, e 2 ) R = 1, (e 1, e 1 ) R = (e 1, e 2 ) R = (e 2, e 3 ) R = (e 3, e 3 ) R = 0. Obviously, L = Re 1 is maximal isotropic but L = Re 1 Re 2 L. 3

4 Proposition 2.6. Any Dirac structure L has a natural Lie algebroid structure together with a representation ρ L = ρ L : L DE on E. Proof. Given a Dirac structure L, by the property (EC-2), we have [X, X] K = 0, for all X Γ(L), which implies that [, ] K L is skew-symmetric. By (4), it follows that (L, [, ] K L, (j ρ) L ) is a Lie algebroid. Moreover, by the property (EC-1), ρ L : L DE is obviously a representation. Remark 2.7. If E is the trivial line bundle M R, then DE = T M E and ρ = a + θ, where a : K T M and θ : K M R. Since ρ is a representation of the Lie algebroid L, where L is a Dirac structure, it follows that θ L = θ L Γ(L ) is a 1-cocycle in the Lie algebroid cohomology associated with L. Therefore, (L, θ L ) is a Jacobi algebroid, which is by definition, a Lie algebroid together with a 1-cocycle in the Lie algebroid cohomology [9]. Please see [22] for general theories of Lie algebroids, Lie algebroid cohomologies and representations. Next we briefly recall the notions of an omni-lie algebroid, a generalized Courant algebroid or a Courant- Jacobi algebroid and a generalized Lie bialgebroid. We will see that the geometric structure of an E- Courant algebroid unifies these structures. In fact, an omni-lie algebroid is one kind of exact E-Courant algebroids and E-Courant algebroids reduce to generalized Courant algebroids if E is a trivial line bundle. Omni-Lie algebroids A natural nondegenerate E-valued pairing, E between JE and DE is given by µ, d E = d, µ E du, µ = [u] m JE, u Γ(E), d DE. Moreover, this pairing is C (M)- linear and satisfies the following properties: µ, Φ E = Φ p(µ), Φ gl(e), µ JE; y, d E = y j(d), y Hom(T M, E), d DE. Furthermore, Γ(JE) is an invariant subspace of the Lie derivative L d for any d Γ(DE), which can be defined by the Leibniz rule as follows: L d µ, d E d µ, d E µ, [d, d ] D E, µ Γ(JE), d Γ(DE). Definition 2.8. [5] The quadruple (E, {, }, (, ) E, ρ) is called an omni-lie algebroid, where E = DE JE, the anchor ρ is the projection from E to DE, {, } and (, ) E are given by (d + µ, r + ν) E 1 2 ( d, ν E + r, µ E ), d, r DE, µ, ν JE, (6) {d + µ, r + ν} [d, r] D + L d ν L r µ + d µ, r E. (7) If there is no risk of confusion, we simply denote the omni-lie algebroid (E, {, }, (, ) E, ρ) by E. The E-valued pairing defined by (6) and the bracket defined by (7) will be referred as the standard pairing and the standard bracket on E = DE JE. About the properties of an omni-lie algebroid, please see [5] for more details. Evidently, ρ = 1 JE, the identity map on JE. Clearly, the omni-lie algebroid E is an E-Courant algebroid. Its Dirac structures are studied in [6] by the authors. Generalized Courant algebroids (Courant-Jacobi algebroids) The notion of a generalized Courant algebroid was first introduced in [24]. It is a pair (K, θ), where K M is a vector bundle equipped with a nondegenerate symmetric bilinear form (, ), a skewsymmetric bracket [, ] on Γ(K) and a bundle map ρ θ : K T M R, which is a first-order differential operator such that some compatibility conditions hold. We may write ρ θ (X) = (ρ(x), θ, X ), where ρ : K T M and θ Γ(K ) = Γ(K) satisfies (θ, [X, Y ]) = ρ(x)(θ, Y ) ρ(y )(θ, X), X, Y Γ(K). We leave the details here and we should note that the skew-symmetric bracket [, ] does not satisfy the Leibniz rule. The notion of a Courant-Jacobi algebroid was introduced in [9]. In [24], the authors 4

5 established the equivalence of a generalized Courant algebroid and a Courant-Jacobi algebroid. Roughly speaking, the difference between them is that the bracket [, ] in a generalized Courant algebroid is skewsymmetric but does not enjoy the Leibniz rule and the operation in a Courant-Jacobi algebroid is not skew-symmetric but enjoy the Leibniz rule. In particular, a generalized Courant algebroid reduces to a Courant algebroid [17] if θ = 0. The bracket [, ] is the skew-symmetrization of the operation and the latter one is known as the Dorfman bracket. Evidently, a generalized Courant algebroid or a Courant-Jacobi algebroid is an E-Courant algebroid, where E = M R and we omit the details here. So it follows that Jacobi algebroids and Courant algebroids are all E-Courant algebroids where E = M R. Generalized Lie bialgebroids A Lie bialgebroid is a pair of vector bundles in duality, each of which is a Lie algebroid, such that the differential defined by one of them on the exterior algebra of its dual is a derivative of the Schouten bracket [16], [23]. A generalized Lie bialgebroid [14] or a Jacobi bialgebroid [8] is a pair ((A, φ 0 ), (A, X 0 )), where A and A are two vector bundles in duality, both of which have Lie algebroid structures (A, [, ], a), (A, [, ], a ) respectively and φ 0 Γ(A ), X 0 Γ(A) are 1-cocycles in the Lie algebroid cohomology such that for all X, Y Γ(A), the following conditions hold: d X0 [X, Y ] = [d X0 X, Y ]φ 0 + [X, d X0 Y ]φ 0, (8) φ 0 (X 0 ) = 0, a(x 0 ) = a (φ 0 ), L φ0 X + L X0 X = 0, (9) where d X0 is the X 0 -differential of A, [, ] φ0 is the φ 0 -Schouten bracket, L and L are the usual Lie derivatives. For more information of these notations, see [14]. For a Jacobi manifold (M, X, Λ), ((T M R, (0, 1)), (T M R, ( X, 0))) is a generalized Lie bialgebroid. Furthermore, for a generalized Lie bialgebroid, there is an induced Jacobi structure on the base manifold M. In particular, both ((A, φ 0 ) and (A, X 0 )) are Jacobi algebroids. If φ 0 = 0 and X 0 =0, a generalized Lie bialgebroid reduces to a Lie bialgebroid. It is known that for a generalized Lie bialgebroid ((A, φ 0 ), (A, X 0 )), there is a natural generalized Courant algebroid which is given by (A A, φ 0 + X 0 ). Next we give more examples of E-Courant algebroids. Example 2.9. Let A be a Lie algebroid and ρ A : A DE a representation of A on a vector bundle E. Let K = A A E, for any X, Y Γ(A), ξ u, η v Γ(A E), we define the following operations: ρ(x + ξ u) = ρ A (X), [X + ξ u, Y + η v] K = [X, Y ] + L X (η v) L Y (ξ u) + ρ A d( Y, ξ u), (X + ξ u, Y + η v) E = 1 ( X, η v + Y, ξ u). 2 Evidently, ρ = ρ A : JE A E and it is straightforward to verify that (A A E, [, ] K, (, ) E, ρ) is an E-Courant algebroid. Example Consider an E-Courant algebroid K whose anchor ρ is zero. Thus, we have ρ = 0 which yields that the bracket [, ] K is skew-symmetric. So K is a bundle of Lie algebras. The property (EC-3) turns out that there is an invariant E-valued pairing. So we get the conclusion that an E-Courant algebroid K whose anchor ρ is zero is the same thing as a bundle of Lie algebras with an invariant E-valued pairing. Example Since the omni-lie algebra is a special omni-lie algebroid whose base manifold is a point, it follows that the omni-lie algebra who is of the form gl(v ) V is an E-Courant algebroid, where E is the vector space V. In fact, for any Lie algebra (g, [, ] g ) with a faithful representation ρ g : g gl(v ) on a vector space V, assume that for any v V, there is some A g such that ρ g (A)(v) 0. Such a representation is called nondegenerate. Then we can introduce a nondegenerate V -valued pairing (, ) V and a bilinear bracket [, ] on the space g V which are given by (A + u, B + v) V = 1 2 (ρ g(a)(v) + ρ g (B)(u)), [A + u, B + v] = [A, B] g + ρ g (A)(v), A + u, B + v g V. 5

6 ρ : g V gl(v ) is defined by ρ(a + u) = ρ g (A) for any A + u g V. Follow from ρ (u)(b + v) = 1 2 ρ g(b)(u) = (u, B) V, (10) we have ρ = 1 V : JV = V V. Clearly, (g V, (, ) V, [, ], ρ) is a V -Courant algebroid. The bracket defined above appeared firstly in [15]. For any representation ρ : g gl(v ), the structure of (g V, [, ]) was called a hemisemidirect product of g with V. There is also a natural exact Courant algebra [2] associated with any g-module. The above example can be generalized to the Lie algebroid case naturally as follows. Example Let (A, [, ], a) be a Lie algebroid with a nondegenerate representation ρ A : A DE. On the vector bundle A JE, define an E-valued pairing (, ) E and a bracket {, } by (X + µ, Y + ν) E = 1 2 ( ρ A(X), ν E + ρ A (Y ), µ E ), {X + µ, Y + ν} = [X, Y ] + L ρ(x) ν L ρ(y ) µ + d ρ A (Y ), µ E, for any X + µ, Y + ν Γ(A JE), and define ρ : A JE DE by ρ(x + µ) = ρ A (X). Similar to (10), we have ρ = 1 JE. Then it is easy to see that (A JE, (, ) E, {, }, ρ) is an E-Courant algebroid. The jet bundle of Courant algebroids At the end of this section, we prove that for any Courant algebroid C, JC is a T M-Courant algebroid naturally. The original definition of a Courant algebroid was introduced in [17]. Here we use the alternative definition given by D. Roytenberg in [25]. Recall that a Courant algebroid is a vector bundle C M together with a nondegenerate bilinear form, on the bundle, a bilinear operation, on Γ(E) and a bundle map a : C T M which satisfies a a = 0 such that (Γ(C),, ) is a Leibniz algebra and some compatibility conditions are satisfied. Next we introduce the T M-valued pairing (, ), the bracket [, ] JC and the anchor ρ : JC D(T M) respectively on the jet bundle JC of the vector bundle C. It turns out that (JC, (, ), [, ] JC, ρ) is a T M-Courant algebroid. For any X, Y Γ(C), the T M-valued pairing (, ) of dx, dy is given by By (3), we can obtain (dx, dy ) = d X, Y. (11) (dx, dg Y ) = X, Y dg, (df X, dg Y ) = 0. For any X, Y Γ(C), the bracket [, ] JC of dx, dy is given by By (4), (5) and (3), we have [dx, dy ] JC = d X, Y. (12) [dx, df Y ] JC = df X, Y + d(a(x)f) Y, [df Y, dx] JC = df Y, X d(a(x)f) Y + 2 X, Y da (df), [df X, dg Y ] JC = a(x)(g)df Y a(y )(f)dg X. For any X Γ(C), ρ(dx) Γ(D(T M)) is given by ρ(dx)( ) = L a(x) ( ). (13) By (3), we can obtain ρ(df X) = a(x) df, f C (M). 6

7 For any ξ Ω 1 (M), we have ρ(dx)(fξ) = L a(x) (fξ) = fl a(x) (ξ) + a(x)(f)ξ, which yields that j ρ dx = a(x), where j : D(T M) T M is the anchor of D(T M) given in (1). Furthermore, for any g C (M), the fact that ρ(df X)(gξ) = gρ(df X)(ξ) yields that j ρ(df X) = 0. We identify C with C by the bilinear form. For any f, g C (M), it is straightforward to obtain the following relations: ρ (ddf) = d(a df), ρ (df dg) = dg a (df), (14) ρ (d(fdg)) = dg a (df) + fd(a dg). Theorem With the above notations, for any Courant algebroid C, (JC, (, ), [, ] JC, ρ) is a T M- Courant algebroid. Proof. It is straightforward to see the pairing (, ) and ρ are bundle maps and (Γ(JC), [, ] JC ) is a Leibniz algebra. Next we verify that the data (JC, (, ), [, ] JC, ρ) satisfies the properties listed in Definition 2.1. It suffices to verify them using the elements dx, dy, dz, df X, dg Y, dh Z, where X, Y, Z Γ(C), f, g, h C (M). First we check the property (EC-1). Clearly, we have ρ[dx, dy ] JC = ρd X, Y = L a X,Y = [L a(x), L a(y ) ] D = [ρdx, ρdy ] D. Furthermore, since a a = 0, we have On the other hand, ρ[df X, dy ] JC = ρ(df X, Y d(a(y )f) X + 2 X, Y da (df)) = a([x, Y ]) df a(x) d(a(y )f). [ρ(df X), ρ(dy )] D (ξ) = [a(x) df, L a(y ) ] D (ξ) = L a(y ) ξ, a(x) df L a(y ) ( a(x), ξ df) which implies Similarly, we have = a([x, Y ]), ξ df a(x), ξ d(a(y )f), ρ[df X, dy ] JC = [ρ(df X), ρ(dy )] D. ρ[dx, df Y ] JC = [ρ(dx), ρ(df Y )] D = a([x, Y ]) df + a(y ) d(a(x)f), ρ[df X, dg Y ] JC = [ρ(df X), ρ(dg Y )] D = (a(x)g)a(y ) df (a(y )f)a(x) dg. To see the property (EC-2), notice that [df X, df X] JC = 0 and (df X, df X) = 0, so we have Furthermore, [df X, df X] JC = ρ d (df X, df X). [dx, dx] JC = d X, X = d a (d X, X ) = ρ d d X, X = ρ d (dx, dx), [df X, dy ] JC + [dy, df X] JC = 2 X, Y ρ ddf + 2df a (d X, Y ) = 2ρ d( X, Y df) = 2ρ d (df X, Y ), which implies that the property (EC-2) holds. It is straightforward to see that the property (EC-3) holds. We leave the details to the readers. The property (EC-4) follows from (14). The property (EC-5) follows from the fact that a a = 0. 7

8 3 The E-dual pair of Lie algebroids To define a class of E-Courant algebroids, called E-Lie bialgebroids, first we discuss E-duality between two Lie algebroids. Let A be a vector bundle and B be a subbundle of Hom(A, E). For any µ k Hom( k A, E), denote by µ k the induced bundle map from k 1 A to Hom(A, E) which is given by µ k (X 1,, X k 1 )(X k ) = µ k (X 1,, X k 1, X k ). (15) We define a series of vector bundles k E B (k 0) by setting 0 E B=E, 1 E B = B and k EB { µ k Hom( k A, E) Im(µ k ) B }, (k 2). (16) Similarly, we have the notion k E A. Assume that (A, [, ], a) is a Lie algebroid and B Hom(A, E) is an E-dual bundle of A, i.e., the E-valued pairing, E : A M B E, X, ξ E ξ(x) X A, ξ B is nondegenerate. Obviously, A is also an E-dual bundle of B meanwhile. A representation ρ A : A DE of A on E is said to be B-invariant if (Γ( E B), da ) is a subcomplex of (Ω (A, E), d A ). If ρ A is a B-invariant representation, we have ρ A (JE) B. In fact, by the definition of ρ A, ρ A(µ)(X) = µ, ρ A (X) E, µ JE, X A, it follows that ρ A : JE B is given by ρ A ([u] m) = (d A u) m, for all u Γ(E). Thus, ρ A (JE) B is equivalent to the condition d A (Γ(E)) Γ(B). Furthermore, for any representation ρ A : A DE and X Γ(A), there are two natural operations of the Lie derivative which is given by L X : Γ(Hom( k A, E)) Γ(Hom( k A, E)) = Γ( k A E), L X (ω u) = (L X ω) u + ω ρ A (X)u, ω Γ( k A ), u Γ(E), and L X : Γ(Hom( k (A E), E)) Γ(Hom( k (A E), E)) = Γ( k (A E ) E), which is given by L X u = ρ A (X)u for all u Γ(E) and k L X Ξ(ϖ 1 ϖ k ) = ρ A (X)(Ξ(ϖ 1 ϖ k )) Ξ(ϖ 1 L X ϖ i ϖ k ), (17) for all Ξ Γ(Hom( k (A E), E)), ϖ i Γ(A E). In particular, since A Hom(A E, E), we have i=1 L X Y = [X, Y ], Y Γ(A). Proposition 3.1. The following statements are equivalent: (1) The representation ρ A : A DE is B-invariant; (2) d A Γ(E) Γ(B) and d A Γ(B) Γ( 2 E B); (3) Γ( k E B) is invariant under the operation of the Lie derivative L X for any X Γ(A); (4) Γ( k E A) is invariant under the operation of the Lie derivative L X for any X Γ(A). 8

9 Proof. The implication of (1) = (2) is obvious. We adopt an inductive approach to see the implication of (2) = (1). For any n 1, L X : Γ( n E B) Γ( n E B) is well defined and we have i Y L X L X i Y = i [Y,X]. Assume that d A Γ( n 1 E B) Γ( n E B) and da Γ( n EB) Γ( n+1 E B) hold, for any µn+1 Γ( n+1 E B), to prove that d A µ n+1 Γ( n+2 E B), it suffices to show that i Xd A µ n+1 Γ( n+1 E B), for all a Γ(A). Again, it suffices to show that i Y i X d A µ n+1 Γ( n EB), for all Y Γ(A). In fact, i Y i X d A µ n+1 = i Y (L X µ n+1 d A i X µ n+1 ) = (i Y L X L X i Y )µ n+1 + L X i Y µ n+1 i Y d A i X µ n+1 = i [Y,X] µ n+1 + L X i Y µ n+1 i Y d A i X µ n+1 Γ( k EB), and we obtain the conclusion that Γ( E B) is a subcomplex of Ω (A, E). This completes the proof of the equivalence of (1) and (2). The equivalence of (1) and (3) is obvious. Next we prove the equivalence of (2) and (4). For any X k Γ( k E A) and any ξ i B, we have iξ1 ξ k 1 L XX k, ξ k E = (L X X k )(ξ 1 ξ 2 ξ k ) = ρ A (X)(X k (ξ 1 ξ 2 ξ k )) = ρ A (X) i ξ1 ξ k 1 X k, ξ k = E k 1 k X k (ξ 1 L X ξ i ξ k ) i=1 iξ1 L X ξ j ξ Xk k 1, ξ k E i ξ1 ξ k 1 X k, L X ξ k j=1 k 1 [X, i ξ1 ξ k 1 X k ] i ξ1 L X ξ j ξ k 1 X k, ξ k j=1 Since the E-valued pairing, E is nondegenerate, we have k 1 i ξ1 ξ k 1 L X X k = [X, i ξ1 ξ k 1 X k ] i ξ1 L X ξ j ξ k 1 X k, which implies the equivalence of (2) and (4). Definition 3.2. For two Lie algebroids A and B, who are E-dual as vector bundles and together with B-invariant representation ρ A : A DE and A-invariant representation ρ B : B DE respectively, we call ((A, ρ A ); (B, ρ B )) an E-dual pair of the Lie algebroids A and B. Since JE and DE are mutual E-dual, the notation k EJE is clear and we call it the k-th skewsymmetric jet vector bundle. Proposition 3.3. If k 2, for any µ k ( k E JE) m, m M, there is a unique bundle map λ µ k Hom( k 1 T m M, E m ) such that µ k (d 1 d k 1 Φ) = Φ λ µ k(j(d 1 ) j(d k 1 )), Φ gl(e m ), d i (DE) m. Proof. By definition, we have µ k (d 1 d k ) = µ k (d 1 d k 1 ), d k, which implies j=1 p µ k (d 1 d k 1 ) = µ k (d 1 d k 1 1 Em ). We claim that i Φ (p µ k ) = 0, for all Φ gl(e m). In fact, we have p µ k (d 1 d k 2 Φ) = µ k (d 1 d k 2 Φ 1 Em ) = µ k (d 1 d k 2 1 Em Φ) = µ k (d 1 d k 2 1 Em ), Φ E = Φ p µk (d 1 d k 2 1 Em ) = Φ µ k (d 1 d k 2 1 Em 1 Em ) = 0. E. E E 9

10 Therefore, the map p µ k : k 1 (DE) m E m factors through j, i.e., there is a unique bundle map λ µ k : k 1 T m M E m such that p µ k (d 1 d k 1 ) = λ µ k(j(d 1 ) j(d k 1 )), which yields the conclusion. Therefore, for k 2, the k-th skew-symmetric jet vector bundle k EJE of E also can be directly expressed by k EJE = {µ k Hom( k DE, E)! λ µ k Hom( k 1 T M, E), s.t., Φ gl(e), d i DE, µ k (d 1 d k 1 Φ) = Φ λ µ k(j(d 1 ) j(d k 1 ))}. We will write λ µ k = p k (µ k ) in this definition, for µ k given above. For any ξ Hom( k T M, E), k 1, we define e k (ξ) k EJE by e k (ξ)(d 1 d k ) ξ(j(d 1 ) j(d k )), d i DE. We regard Hom( 1 T M, E) = 0, Hom( 0 T M, E) = E and 0 E JE = E. Let p0 = 0 and e 0 = 1 E. We have the following conclusion. Proposition 3.4. For any k 0, the following sequence is exact: 0 Hom( k T M, E) ek k EJE pk Hom( k 1 T M, E) 0. (18) Proof. If k = 0, 1, the result is clear. For k 2, e k is an injection and p k e k = 0 are also evident. Now we prove that p k is surjective. For any λ Hom( k 1 T m M, E m ), we define a map λ Hom(( k DE) m, E m ) by λ(d 1 d k ) k ( 1) i+1 d i λ(j(d 1 ) ĵ(d i) j(d k )), d i (DE) m. i=1 For any Φ gl(e m ), it is evident that λ(d 1 d k 1 Φ) = ( 1) k+1 Φ λ(j(d 1 ) j(d k 1 )). Therefore, λ ( k E JE) m and p k λ = ( 1) k+1 λ. Finally, if µ k ( k E JE) m satisfies p k (µ k ) = 0, then we have µ k (d 1 d k 1 Φ) = Φ p k (µ k )(j(d 1 ) j(d k 1 )) = 0, which implies that the map µ k factors through j, i.e., there is a unique ξ Hom( k T m M, E m ) such that Therefore, sequence (18) is exact. µ k (d 1 d k ) = ξ(j(d 1 ) j(d k )). Sequence (18) will be called the k-th skew-symmetric jet sequence of E. Again, in most cases, we will omit the embedding e k and directly regard Hom( k T M, E) k EJE. Evidently, we have p(i d µ k ) = i j(d) p k µ k, d DE, µ k k EJE. (19) Recall that DE is a Lie algebroid and there is a natural representation 1 DE on E. For µ k Γ( k E JE) and d i Γ(DE), the coboundary operator d : Ω (DE, E) Ω +1 (DE, E) is given by dµ k (d 1 d k+1 ) k+1 ( 1) i+1 d i µ k (d 1 d i d k+1 ) i=1 + i<j( 1) i+j µ k ([d i, d j ] D d 1 d i d j d k+1 ). (20) 10

11 Lemma 3.5. With the above notations, the representation 1 DE is JE-invariant, i.e., (Γ( EJE), d) is a subcomplex of (Γ(Hom( DE, E)), d). More precisely, for any d Γ(DE) and µ k Γ( k EJE), k 0, we have p k (L d µ k ) = L d (p k (µ k )), (21) p k dp k µ k = ( 1) k+1 p k µ k, (22) p k+1 dµ k = dp k µ k + ( 1) k µ k. (23) Proof. Assume that p k µ k = λ Γ(Hom( k 1 T M, E)), i.e., for all Φ Γ(gl(E)), d i Γ(DE), By straightforward computations, we have µ k (d 1 d k 1 Φ) = Φ λ(j(d 1 ) j(d k 1 )). L d µ k (d 1 d k 1 Φ) = d µ k (d 1 d k 1 Φ) µ k (d 1 d k 1 [d, Φ]) k 1 µ k (d 1 [d, d i ] D d k 1 Φ) i=1 = d Φ λ(jd 1 jd k 1 ) [d, Φ] λ(jd 1 jd k 1 ) Φ λ(jd 1 [jd, jd i ] jd k 1 ) k 1 i=1 k 1 = Φ(d λ(jd 1 jd k 1 ) λ(jd 1 [jd, jd i ] jd k 1 )) = Φ(L d λ(d 1 d k 1 )), i=1 which implies that we have the equality p k (L d µ k ) = L d (p k (µ k )). The other conclusions can be obtained similarly. Remark 3.6. In fact, for any λ Γ(Hom( k 1 T M, E)) Γ( k 1 E JE), L dλ also means the Lie derivative of Ω k 1 (M) Γ(E) in the obvious sense: L d L j(d) 1 Γ(E) + 1 Ω k 1 (M) d. By Lemma 3.5, the representation 1 DE of DE is JE-invariant. Furthermore, JE is a Lie algebroid with all the structures zero. Thus we have Corollary 3.7. ((DE, 1 DE ); (JE, 0)) is an E-dual pair. Theorem 3.8. For the cochain complex C(E) = (Γ( E JE), d), we have Hk (C(E)) = 0, for all k = 0, 1, 2,. In other words, there is a long exact sequence: where n = dimm Γ(E) d Γ(JE) d Γ( 2 EJE) d d Γ( n EJE) 0, (24) Proof. If k = dimm + 2, we have Hom( k+2 T M, E) = Hom( k+1 T M, E) = 0. By the k-th skewsymmetric jet sequence (18), it is evident that k+2 E JE = 0. By (23), if dµk = 0, we have µ k = ( 1) k+1 d(p k µ k ), which implies that H k (C(E)) = 0. Next we give two examples where E is a vector space and a trivial line bundle M R respectively to see how we get the exact sequence (24). 11

12 Example 3.9. If E reduces to a vector space V, JV = V, DV = gl(v ). Since M reduces to a single point, by the k-th skew-symmetric jet sequence (18), we have 2 V V = 0. In fact, for any φ 2 V V, A, B gl(v ), we have φ(a B) = Bφ(A 1 V ) = BAφ(1 V 1 V ) = 0. Therefore, φ = 0 and 2 V V = 0. Example Let E = M R, then we have DE = T M R and JE = T M R. Denote the basis of C (M) Γ(DE) by 1 and the dual basis by 1, i.e., 1 (f1) = f, for all f C (M). Since ρ : DE JE DE is the projection, we can write ρ = pr T M + 1, where 1 is considered to be a section of E satisfying 1 (f) = f, 1 (X) = 1 (µ) = 0, for all f C (M), X X(M) and µ Γ(JE). Since df = df + f1, we have df = 0 f = 0, f C (M), which implies that H 0 (C(E)) = 0. For any λ Ω 1 (M), by (23), we have p 2 (dλ) = λ. Furthermore, we have d(f1 ) = d(d(f) df) = d(df), which implies p 2 (d(f1 )) = df. Therefore, for any µ Γ(JE), we have dµ = 0 µ = df + f1 µ = df, for some f C (M), which implies that H 1 (C(E)) = 0. Similarly, we have H k (C(E)) = 0. 4 The automorphism group of omni-lie algebroids As two direct applications of Theorem 3.8, the automorphism group and the twist of the omni-lie algebroid are fixed in this section. It is subtle to define morphisms between two E i -Courant algebroids (i = 1, 2) with different base manifolds, which will be studied in the future. Please refer to [4], [12] for Lie algebroid morphisms and [3] for Courant algebroid morphisms. Certainly, it is clear to define isomorphisms for E-Courant algebroids. Here we only consider a special case. For any automorphism Φ : E E, denote by φ the induced diffeomorphism on the base manifold M. There is a unique induced automorphism Ad Φ of DE which is given by Ad Φ (d)(u) = Φ d Φ 1 (u), d Γ(DE), u Γ(E). Definition 4.1. The automorphism group Aut(K) of an E-Courant algebroid K is the group of bundle automorphisms F : K K covering bundle automorphisms Φ : E E such that 1). (F (X), F (Y )) E = Φ (X, Y ) E, i.e., F is orthogonal, 2). F [X, Y ] K = [F (X), F (Y )] K, i.e., F is bracket-preserving, 3). ρ F = Ad Φ ρ, i.e., F is compatible with the anchor. It is easy to see that the set of all automorphisms (F, I E ) is a normal subgroup of Aut(K), which is similar to the B-field introduced in [10]. Now let us study the automorphism group of the omni-lie algebroid E = DE JE defined in Definition 2.8. For any automorphism Φ : E E, there is an induced map Φ : JE JE which is given by Φ(µ) = [Φ(u)] φ(m), µ = [u] m JE m, u Γ(E). It is clear that the pair (Φ, Ad Φ + Φ) is an automorphism of the omni-lie algebroid E and it is totally determined by the automorphism Φ of the vector bundle E. Meanwhile, there is another symmetry of the omni-lie algebroid E, which we call the B-field transformation. For any b Γ(Hom(DE, JE), there is a transformation e b : E E which is defined by d + µ d + µ + i d b, d + µ E. 12

13 Lemma 4.2. For any b Γ( 2 E JE), the map eb is an automorphism of the omni-lie algebroid E if and only if b is closed, i.e., db = 0. Proof. Let d, r Γ(DE), µ, ν Γ(JE) and b Γ( 2 EJE). First, b is skew-symmetric yields that eb preserves the standard pairing given in (6). Then, {e b (d + µ), e b (r + ν)} = {d + µ, r + ν} + {d, i r b} + {i d b, r} = {d + µ, r + ν} + L d i r b i r di d b = e b ({d + µ, r + ν}) + i r i d db. (25) Therefore, e b is an automorphism of the omni-lie algebroid E if and only if i r i d db = 0 for all d, r Γ(DE), which happens precisely when db = 0. For any b Γ( 2 E JE), if db = 0, we call eb the B-field transformation. Corollary 4.3. The group of the B-field transformations is isomorphic to Γ(Hom(T M, E)). Proof. For any b Γ( 2 E JE) such that db = 0, assume µ = p2 b, where p k is given in sequence (18). By (23), b = dµ. As a vector space, Γ(JE) = Γ(Hom(T M, E)) Γ(E). Any u Γ(E), du Γ(JE). Therefore, it is evident that d(γ(je)) = d(γ(hom(t M, E))) = Γ(Hom(T M, E)). In fact, any automorphism of the omni-lie algebroid E is a composition of an automorphism Φ of the vector bundle E and a B-field transformation. Theorem 4.4. Let (Φ, F ) be an automorphism of the omni-lie algebroid E, where Φ is an automorphism of E and F : E E is an automorphism of E. Then F must be a composition of an automorphism Φ of the vector bundle E and a B-field transformation e b. To be more precise, the automorphism group Aut(E) is the semidirect product of Aut(E) and Γ(Hom(T M, E)), i.e., Aut(E) = Aut(E) Γ(Hom(T M, E)). (26) Proof. Since Φ is an automorphism of E, (Φ, Ad Φ + Φ) is an automorphism of the omni-lie algebroid E. Setting G = Φ 1 F, the pair (1 E, G) is also an automorphism of the omni-lie algebroid E. Since G respects ρ, we can write G(d + µ) = d + b(d) + σ(µ), d + µ E, where b : DE JE and σ : JE JE are two bundle maps. Then by (G(d + µ), G(r + ν)) E = (d + µ, r + ν) E, d + µ, r + ν E, we have σ = 1 JE and b is skew-symmetric: b(d), r E = b(r), d E. d, r DE. Follow from the equation {G(d + µ), G(r + ν)} = G{d + µ, r + ν}, d + µ, r + ν Γ(E), we have that b is closed with respect to the Lie algebroid cohomology of DE. Hence we have F = Φ G, where G = e b, in which b is closed, as required. Differentiating a 1-parameter family of automorphisms F t = Φ t e tb, F 0 = 1, b = dµ, we see that the Lie algebra Der(E) of infinitesimal symmetries of the omni-lie algebroid E consists of the pairs (d, µ) Γ(DE) Γ(Hom(T M, E)), which act via (d + µ) (r + ν) = [d, r] D + L d ν i r dµ, r + ν Γ(E). (27) Follow from Theorem 4.4, we can conclude the following 13

14 Proposition 4.5. The Lie algebra Der(E) of infinitesimal symmetries of an omni-lie algebroid is isomorphic to the semidirect sum of Γ(DE) and Γ(Hom(T M, E)), i.e., Der(E) = Γ(DE) Γ(Hom(T M, E)). Furthermore, all of these derivations are defined by the standard bracket (7) of the omni-lie algebroid E from the left side, and there is an exact sequence of Leibniz algebra morphism: 0 Γ(E) d Γ(E) ad Der(E) 0. (28) Proof. By (27), for any derivation d + µ, we have (d + µ) (r + ν) = {d + µ, r + ν}, which implies that the derivations of the omni-lie algebroid E are defined by the standard bracket (7). Since d 2 = 0, we have that d(γ(e)) is the left center of the standard bracket (7), i.e., the kernel of the map ad. Thus, we obtain the exact sequence (28). Notice that the subbundle DE Hom(T M, E) of E with the bracket on its section space Der(E) is only a local Lie algebra instead of a Lie algebroid since the action of DE on Hom(T M, E) by the Lie derivative is not a Lie algebroid representation. Similar as the Courant algebroid can be twisted by a closed 3-form, now we consider the deformation of the omni-lie algebroid E in the following way: given a Θ : Γ(DE DE) Γ(JE), define a new bracket {, } Θ on Γ(DE JE) by {d + µ, r + ν} Θ = {d + µ, r + ν} + Θ(d r). By the property (EC-2), Θ is skew-symmetric from Γ(DE) Γ(DE) to Γ(JE). By the property (EC-3), we have Thus, 0 = ([d, r] K, t) E + (r, [d, t] K ) E = 1 2 ( Θ(d r), t E + Θ(d t), r E ) = 1 2 ( Θ(d r), t E Θ(t d), r E ). Θ(d r), t E = Θ(r t), d E = Θ(t d), r E, d, r, t DE, (29) which yields that Θ Γ( 3 E JE). Furthermore, we can easily deduce that {, } Θ defines an E-Courant algebroid structure on DE JE (using the standard pairing (6) and the same anchor) if and only if dθ = 0. We call this E-Courant algebroid the twisted omni-lie algebroid. Theorem 4.6. Any twisted omni-lie algebroid is isomorphic to the omni-lie algebroid E. Proof. By Theorem 3.8, there is some b Γ( 2 EJE) such that Θ = db. By (25), we have e b {d + µ, r + ν} Θ = {d + µ, r + ν}, d + µ, r + ν Γ(E). Furthermore, b is skew-symmetric yields that e b preserves the standard pairing (6). Therefore, the transformation e b is an isomorphism. 5 Exact E-Courant algebroids Definition 5.1. An E-Courant algebroid (K, (, ) E, [, ] K, ρ) is said to be exact if the following sequence is exact: ρ 0 ρ JE K DE 0. (30) 14

15 Obviously, the omni-lie algebroid is an exact E-Courant algebroid. In [28], P. Ševera and A. Weinstein found that any exact Courant algebroid structure on T M T M is a twist of the standard one by a closed 3-form. An important ingredient in the proof is the fact that any exact Courant algebroid has an isotropic splitting, i.e., both T M and T M are isotropic subbundles. Unfortunately, such a fact is not always true for an exact E-Courant algebroid if rank(e) 2. Therefore, we need the theory of the Leibniz algebra cohomology for general cases. In the following, we will prove any bracket [, ] K on an exact E-Courant algebroid DE JE is a twist of the standard bracket (7) by a 2-cocycle in the Leibniz cohomology of Γ(DE) with coefficients in Γ(JE), which is also a 3-cocycle in the Leibniz cohomology of Γ(DE) with coefficients in Γ(E). So it is a generalization from derham cohomology (commutative) to Leibniz cohomology (noncommutative). Recall that a representation of the Leibniz algebra (g, [, ]) is an R-module V equipped with two actions (left and right) of g, such that [, ] : g V V, [, ] : V g V, l [g1,g 2] = [l g1, l g2 ], r [g1,g 2] = [l g1, r g2 ], r g2 l g1 = r g2 r g1, where l g1 g = [g 1, g] and r g1 g = [g, g 1 ]. The Leibniz cohomology of g with coefficients in V is the homology of the cochain complex C k (g, V ) = Hom R ( k g, A), (k 0) whose coboundary operator : C k (g, V ) C k+1 (g, V ) is defined by c k (g 1,, g k+1 ) = k ( 1) i+1 g i ((c k (g 1,, ĝ i,, g k+1 )) + ( 1) k+1 (c k (g 1,, g k ))g k+1 i=1 + ( 1) i c k (g 1,, ĝ i,, g j 1, [g i, g j ], g j+1,, g k+1 ). 1 i<j k+1 The condition = 0 was proved in [20]. For the omni-lie algebroid E, since Γ(E) ia a Leibniz algebra [5] and Γ(DE) is a Lie algebra, there are two actions of the Leibniz algebra Γ(DE) on Γ(JE) which are given by {d, ν} = L d ν, {µ, r} = L r µ + d µ, r E d, r Γ(DE), µ, ν Γ(JE). (31) For any b Γ(Hom(DE, JE)), b = 0 means that L d b(r) L r b(d) + d b(d), r E b([d, r] D ) = 0. For any Θ Γ(Hom( 2 DE, JE)), Θ = 0 means that L d Θ(r, t) L r Θ(d, t) + L t Θ(d, r) d t, Θ(d, r) E + Θ(d, [r, t] D ) Θ(r, [d, t] D ) Θ([d, r] D, t) = 0. (32) Furthermore, obviously, there are two actions of the Leibniz algebra Γ(DE) on Γ(E), which are given by [d, u] = du, [v, r] = rv d, r Γ(DE), u, v Γ(E). (33) Any two chain Θ Γ(Hom( 2 DE, JE)) can be considered as a 3-chain Θ Γ(Hom( 3 DE, E)): The following lemma is straightforward. Θ(d r t) = Θ(d r), t E. Lemma 5.2. With the above notations, if Θ is closed, so is Θ. Definition 5.3. A pair (ω, Θ), where ω : DE DE E is a symmetric bundle map and Θ Γ(Hom( 2 DE, JE)) is called a admissible pair of the omni-lie algebroid E if the following conditions are satisfied: 15

16 1). Θ is a 2-cocycle in the Leibniz cohomology of Γ(DE) with coefficients in Γ(JE), 2). For any d Γ(DE), Θ(d d) = d(ω(d d)), 3). For any d, r Γ(DE), 1 2 dω(r r) = (Θ(d r), r) E + ω([d, r] D r). Two admissible pairs (ω, Θ) and ( ω, Θ) are said to be equivalent if there is some b Γ(Hom(DE, JE)) such that 1). For any d, r Γ(DE), ω(d r) = ω(d r) ( b(d), r E + b(r), d E ), 2). Θ = Θ + b. Clearly, for any b Γ(Hom(DE, JE)), b is a 2-cocycle and ω b : DE DE E which is defined by ω b (d r) = 1 2 ( b(d), r E + b(r), d E ), is a symmetric bundle map. It is straightforward to see (ω b, b) is an admissible pair of the omni-lie algebroid E which is equivalent to the admissible pair (0, 0). Theorem 5.4. There is a one-to-one correspondence between the isomorphic classes of exact E-Courant algebroids and equivalence classes of admissible pairs of the omni-lie algebroid E. More precisely, for any exact E-Courant algebroid (K, (, ) E, [, ] K, ρ), K = DE JE as vector bundles and ρ is the projection to DE, there is an admissible pair (ω, Θ) such that (d + µ, r + ν) E = 1 2 ( d, ν E + r, µ E ) + ω(d r), d + µ, r + ν Γ(DE JE), (34) [d + µ, r + ν] K = {d + µ, r + ν} Θ = {d + µ, r + ν} + Θ(d r), (35) where {, } is the standard bracket given in (7). Conversely, for any admissible pair (ω, Θ), (DE JE, (, ) E, [, ] K, ρ) is an exact E-Courant algebroid, where (, ) E and [, ] K are given in (34) and (35). Furthermore, two exact E-Courant algebroids corresponding to the equivalent admissible pairs are isomorphic. Proof. We split the proof into four steps. In Step (1), we prove that the E-valued pairing of an exact E-Courant algebroid K is of the form (34). In Step (2), we prove that the bracket of an exact E-Courant K is of the form (35). Since the pair (ω, Θ) comes from the exact E-Courant algebroid K, it follows that it is an admissible pair. In Step (3), we prove that if we choose different splitting, we obtain equivalent admissible pairs. In Step (4), we give the proof of the reverse statement. Step (1). As a vector bundle, K = DE JE is evident by choosing a splitting s : DE K of the exact sequence (30) and ρ is transferred to the projection from E to DE. By properties (EC-4) and (EC-5), for all µ, ν JE, we have (ρ µ, ρ ν) E = 1 2 µ, ρ ρ ν E = 0, (36) which implies ρ JE is isotropic under the pairing (, ) E. So if we transfer the pairing (, ) E on K to a pairing (, ) E on DE JE, JE is isotropic. For any d DE, we have Furthermore, for all d, r DE, we have (d, ν) E = (s(d), ρ (ν)) E = 1 2 d, ν E. (37) (d, r) E = (s(d), s(r)) E ω(d r), (38) where ω : DE DE E is a symmetric bundle map. By (36), (37) and (38), it follows that the pairing is of the form (34). 16

17 Step (2). For any d, r Γ(DE), by properties (EC-1) and (EC-2), we may suppose that [d, r] K = [d, r] D + Θ(d r), (39) where Θ is a R-linear mapping Γ(DE) Γ(DE) Γ(JE). By (4), we know that Θ is also C (M)-linear and hence Θ Γ(Hom( 2 DE, JE)). Again by the property (EC-1), there is a bi-linear map such that : Γ(DE) Γ(JE) Γ(JE), [d, µ] K = (d, µ), By the property (EC-3) and JE is isotropic, we have d Γ(DE), µ Γ(JE). d (r, µ) E = ([d, r] K, µ) E + (r, [d, µ] K ) E = ([d, r] D, µ) E + (r, (d, µ)) E, which implies that (d, µ) = L d µ, i.e., Again by the property (EC-3), we have [d, µ] K = L d µ, d Γ(DE), µ Γ(JE). (40) ([µ, ν] K, d) E + (ν, [µ, d] K ) E = 0, µ, ν Γ(JE), d Γ(DE), which implies Furthermore, we have [µ, ν] K = 0, µ, ν Γ(JE). (41) Therefore, d d, µ E = d (d + µ, d + µ) E d (d, d) E = [d + µ, d + µ] K d (d, d) E = [d, µ] K + [µ, d] K = L d µ + [µ, d] K. By (39), (40), (41) and (42), we get [µ, d] K = d d, µ E L d µ, µ Γ(JE), d Γ(DE). (42) [d + µ, r + ν] K = {d + µ, r + ν} + Θ(d r), d + µ, r + ν Γ(E). (43) Furthermore, since the bracket [, ] K satisfies the Leibniz rule, we have L d Θ(r, t) L r Θ(d, t) + L t Θ(d, r) d t, Θ(d, r) E + Θ(d, [r, t] D ) Θ(r, [d, t] D ) Θ([d, r] D, t) = 0, which implies Θ is a 2-cocycle in the Leibniz cohomology of Γ(DE) with coefficients in Γ(JE) and the two actions are given in (31). Since the pair (ω, Θ) comes from the E-Courant algebroid K, it is straightforward to see that it is an admissible pair. Step (3). Suppose that we have two sections s 1, s 2 : DE K, then we have ρ(s 1 s 2 ) = 0. So consider b = s 1 s 2 : DE JE, in the s 1 splitting, we have s 2 (d) = d + b(d). By straightforward computations, we have {d + b(d), r + b(r)} Θ = [d, r] D + L d b(r) i r b(d) + Θ(d r) = [d, r] D + b([d, r] D ) + (Θ + b)(d r). (44) In particular, in the splitting of s 2, Θ changes by b. We should note that the induced pairing on DE also changes. If we denote the new pairing by ω, for any d, r Γ(DE), we have ω(d, r) = ω(d, r) + (b(d), r) E + (b(r), d) E. 17

18 Therefore, if we choose different splitting, we obtain equivalent admissible pairs. Step (4). Conversely, for any admissible pair (ω, Θ), on DE JE, we define the pairing (, ) E and the bracket {, } Θ by (34) and (35). It is straightforward to see that (DE JE, (, ) E, {, } Θ, ρ) is an E-Courant algebroid. If we choose different representative element ( ω, Θ), assume that Θ = Θ + b for some b Γ(Hom(DE, JE)) and the corresponding E-Courant algebroid is (DE JE, (, ) E, {, } Θ+ b, ρ). By straightforward computations similar as (44), for any d + µ, r + ν Γ(DE JE), we have e b {d + µ, r + ν} Θ+ b = {e b (d + µ), e b (r + ν)} Θ. Furthermore, it is evident that ( e b (d + µ), e b (r + ν) ) E = (d + µ, r + ν) E. Therefore, the transformation e b JE, (, ) E, {, } Θ, ρ). is the isomorphism from (DE JE, (, ) E, {, } Θ+ b, ρ) to (DE Remark 5.5. The case where the induced symmetric bundle map ω : DE DE E is zero, i.e., the splitting is isotropic, has been studied at the end of Section 3, which is the twisted omni-lie algebroid. In some special cases, we can define an isotropic splitting as follows. Proposition 5.6. With the above notations, consider the induced E-valued pairing ω : DE DE E on DE, if Im(ω ) JE, we can define an isotropic splitting by setting s(d) = 1 2 ω (d). In particular, if E is a line bundle, there is always an isotropic splitting. Proof. If Im(ω ) JE, consider the splitting s which is given by s(d) = 1 2 ω (d). For any d, r DE and µ, ν JE, we have (d + s(d), r + s(r)) E = (d, r) E ω(d, r) = 0. Thus, s(de) is isotropic, as desired. Furthermore, for any Φ gl(e), Im(ω ) JE ω(d, Φ) = Φ(ω(d, 1 E )). So the conclusion follows from if E is a line bundle, gl(e) is a trivial line bundle. 6 E-Lie bialgebroids In this section we study a class of E-Courant algebroids, called E-Lie bialgebroids. The most notations used below are given already in Section 3. Definition 6.1. An E-dual pair ((A, ρ A ); (B, ρ B )) is called an E-Lie bialgebroid if for all X, Y Γ(A), u, v Γ(E), the following conditions are satisfied. (1) d B [X, Y ] = L X (d B Y ) L Y (d B X), (2) L d A ux = L d B ux, (3) d B u, d A u E = 0. Remark 6.2. Condition (3) is equivalent to ρ B d A = ρ A d B. When there is no confusion we simply denote the E-Lie bialgebroid by (A, B). Recall the definition of the omni-lie algebroid associated with a vector bundle E, it is evident that (DE, JE) is a natural example of E-Lie bialgebroids, where ρ JE = 0 and ρ DE is the identity map. For a Lie bialgebroid (A, A ), it is an E-Lie bialgebroid, where E = M R, the trivial line bundle. The representations ρ A and ρ A are the anchors of A and A respectively. For a Lie algebroid A and a representation ρ A : A DE, (A, A E) is an E-Lie bialgebroid, where ρ A E = 0 (Example 2.9). 18

19 Proposition 6.3. A generalized Lie bialgebroid ((A, φ 0 ), (A, X 0 )) is an E-Lie bialgebroid, where E = M R, a trivial line bundle. Proof. For any X Γ(A), ξ Γ(A ), the representations ρ A and ρ A are given by ρ A (X) = a(x) + φ 0 (X), ρ A (ξ) = a (ξ) + X 0 (ξ), where a and a are the anchors of A and A respectively. Evidently, we have d A = d X0. Furthermore, by definition, we have [X, d X0 Y ] φ0 = [X, d X0 Y ] φ 0, X d X0 Y = L X d X0 Y, X, Y Γ(A). (45) Since ((A, φ 0 ), (A, X 0 )) is a generalized Lie bialgebroid, we have (8). By (45), condition (1) in Definition 6.1 holds, i.e., we have d A [X, Y ] = L X (d A Y ) L Y (d A X). (46) To prove that condition (2) in Definition 6.1 holds, we substitute Y by fy in (46), where f C (M), which yields that L d A f X = L d A f X + f(l φ0 X + L X0 X). By (9), we have L da f X = L d A f X, (47) which is condition (2) in Definition 6.1. At last, substitute X by fx in (47), we have ρ A d A (f) = ρ A d A (f) + (a(x 0 ) + a (φ 0 ))(X). By (9), we have ρ A d A (f) = ρ A d A (f), which is condition (3) in Definition 6.1. Therefore, a generalized Lie bialgebroid ((A, φ 0 ), (A, X 0 )) is an E-Lie bialgebroid. Let (A, [, ], a) be a Lie algebroid and ρ A : A DE is a B-invariant representation, where B is an E-dual bundle of A. For any u Γ(E), X Γ(A) and X k Γ( k EA), we can define their Schouten brackets by [u, X k ] = [X k, u] = ( 1) k+1 i d A ux k, [X, X k ] = [X k, X] = L X X k. The Schouten bracket of H, K Γ( 2 E A) is the trilinear mapping [H, K] Γ( 3 EA) defined by the formula [H, K](ξ 1, ξ 2, ξ 3 ) = L Kξ1 ξ 2, Hξ 3 E + L Hξ1 ξ 2, Kξ 3 E + c.p., ξ i Γ(B). (48) In fact, for any Λ Γ( 2 E A), we introduce a bracket [, ] Λ on Γ(B) by [ξ, η] Λ = L Λξ η L Λη ξ d A (Λ(ξ, η)), ξ, η Γ(B). (49) By straightforward computations, we have the following formula [Hξ, Hη] = H[ξ, η] H + 1 [H, H](ξ, η), (50) 2 which implies that [H, H] Γ( 3 E A). [H, K] Γ( 3 EA) follows from replacing H by H + K. For any Λ Γ( 2 E A), let Λ : B A and [Λ, Λ] : 2 B A be the induced bundle maps defined by (15). Moreover, we denote by the compositions respectively. a B = a Λ : B T M and ρ B = ρ A Λ : B DE Proposition 6.4. With the above notations, then (B, [, ] Λ, a B ) is a Lie algebroid together with an A- invariant representation ρ B iff 19

20 (1) ρ A [Λ, Λ] = 0; (2) L X [Λ, Λ] = 0, X Γ(A). Proof. By (50), for any ξ, η Γ(B), we have ρ B ([ξ, η] Λ ) = ρ A Λ ([ξ, η] Λ ) = [ρ B ξ, ρ B η] 1 2 ρ A [Λ, Λ] (ξ, η). (51) Therefore, ρ B is a representation iff ρ A [Λ, Λ] = 0. It is simple to see that for any f C (M), Furthermore, by (50), we have [ξ, fη] Λ = a B (ξ)(f)η + f[ξ, η] Λ. a B ([ξ, η] Λ ) = [a B ξ, a B η] 1 2 j ρ A [Λ, Λ] (ξ, η). Therefore, if ρ A [Λ, Λ] = 0, a B is a homomorphism. Next we consider the condition of the Jacobi identity. Let J(ξ 1, ξ 2, ξ 3 ) = [[ξ 1, ξ 2 ] Λ, ξ 3 ] Λ + c.p., ξ i Γ(B). For any X Γ(A), under the condition ρ A [Λ, Λ] = 0, by straightforward computations [18], we have J(ξ 1, ξ 2, ξ 3 ), X E = { 1 2 [X, Λ (ξ 1, ξ 2 )], ξ 3 E + c.p.} + ρ A (X)([Λ, Λ](ξ 1, ξ 2, ξ 3 )) = 1 2 (L X[Λ, Λ])(ξ 1, ξ 2, ξ 3 ). Thus, under the condition ρ A [Λ, Λ] = 0, the bracket [, ] Λ defined by (49) satisfies the Jacobi identity iff L X [Λ, Λ] = 0 for all X Γ(A). At last we show that the representation ρ B of the Lie algebroid (B, [, ] Λ, a B ) is A-invariant. In fact, by straightforward computations, we have d B u = i d A uλ, d B X = L X Λ, u Γ(E), X Γ(A), (52) which implies that the representation ρ B is A-invariant. By (52), it is straightforward to see that the compatibility conditions of an E-Lie bialgebroid are satisfied. Thus we have Theorem 6.5. With the same assumptions as in Proposition 6.4, (A, B) is an E-Lie bialgebroid. If (A, A ) is a Lie bialgebroid, the base manifold M enjoys a Poisson structure. If (A, A ) is a generalized Lie bialgebroid, the base manifold M enjoys a Jacobi structure. For an E-Lie bialgebroid (A, B) defined in Definition 6.1, we introduce a bracket [, ] E on Γ(E) as follows [u, v] E d B u, d A v E ( = ρ A(d B u)v ), u, v Γ(E). (53) Theorem 6.6. Let (A, B) be an E-Lie bialgebroid, 1) If rank(e) 2, (E, [, ] E ) is a Lie algebroid with the anchor j ρ A ρ B d. 2) If ranke = 1, (E, [, ] E ) is a local Lie algebra. Proof. By condition (3) in Definition 6.1, we know that the bracket defined by (53) is skew-symmetric. To check the Jacobi identity, for all u, v, w Γ(E), we have [u, [v, w] E ] E = d A u, d B d A w, d B v E = d A u, d B (i E d A wd B v) + i d A wd B d B v E = d A u, L da wd B v E = d A u, [d B v, d B w] E = ρ A([d B v, d B w])u = ρ A (d B v)ρ A (d B w)u + ρ A (d B w)ρ A (d B v)u = [v, [w, u] E ] E + [w, [v, u] E ] E. 20